Information is essential to the development of human society and is the primary means by which humans can comprehend society and improve their cognition. In general, the effects of various types of information on society vary. Information that is beneficial to social development, such as knowledge, innovation, and affinity, must be disseminated [1]; [2]; [3]; [4], while information that is harmful to social development, such as rumors and computer viruses, must be contained [5]; [6]; [7]. Therefore, it is valuable to investigate the characteristics and mechanisms of information dissemination.

Information dissemination is similar to the spread of infectious diseases. Daley and Kendall [8] first applied the classical infectious disease model to the spread of rumors in order to study the spread of harmful information, and they proposed the classical rumor spread DK model. The SI model [9], the SIS model [10], the SIR model [11], and other infectious disease models were then used as a research foundation to propose the SEIR model [12], the SPNR model [13], the ILSR model [14], etc.

The aforementioned models typically regard environmental factors as deterministic factors and do not account for them. However, there are many uncertain environmental factors in the real social environment. Random factors, such as government regulation of the media, sudden mass events, and outbreaks of epidemics or geological disasters, can interfere with the dissemination of information [15]. Therefore, researchers are beginning to consider the impact of uncertain environmental factors on the spread of information.

In recent years, extensive studies on stochastic factors in the field of virus propagation have been proposed. Shang [16] developed a low-dimensional system of nonlinear ordinary differential equations to model the mixed susceptible–infected (–recovered) SI(R) epidemics on a random network with general degree distributions. Zhang and Li [17] developed the stochastic SEIR model with migration and human awareness in complex networks and compared the distance between stochastic and deterministic systems at points of local disease equilibrium. Liu and Jiang [18] developed the stochastic SEIR model with asymptomatic infected individuals and calculated the specific form of the probability density near the quasi-endemic equilibrium of the stochastic system. Hussain et al. [19] proposed the stochastic SIRV model with general nonlinear incidence and vaccination and argued that stochastic fluctuations could limit the spread of disease. Wang et al. [20] designed the stochastic SICA model with standard incidence rates, provided sufficient conditions for HIV extinction and persistence, and discovered that increasing the intensity of stochastic perturbations could control the virus’s spread. Bobryk [21] believed that the spread rate in the SIR model must be positive, so he incorporated telegraphic noise, trichotomous noise, and bounded noise into the stochastic model and then discussed the effect of the stochastic perturbation on the stability behavior of disease-free equilibrium points. Kiouach and Sabbar [22] used Gaussian white noise and Lévy jumps to represent, respectively, parametric perturbations in stochastic models, and they calculated the disease’s mean persistence, ergodicity, and extinction. In addition, the application of infectious disease models with stochastic parameter perturbations to the novel coronavirus has been a hot topic over the past few years [23]; [24]; [25].

In the field of information spread, researchers have focused on the impact of stochastic factors on models of rumor spread. Cheng et al. [26] argued that individual activities play a significant role in rumor spread and that individual activities are affected by many uncertainties. As a result, they constructed the I _a I _d S _a S _d R _a R _d model and analyzed the dynamical behavior of the stochastic model based on this argument. Zhang and Zhu [27] concluded that the spread rate among individuals in rumor spread models is usually disturbed by random factors, and increasing the noise intensity can effectively control the spread of rumors. Tong et al. [28] constructed a stochastic rumor spread model with media coverage and age structure based on a deterministic model and confirmed the existence of a global positive solution to the model and the necessary conditions for rumor disappearance and smooth distribution. Zhou et al. [29] argued that information intervention influences rumor spread; consequently, they developed a stochastic SIRZ rumor spread model considering the information intervention mechanism and proposed an optimal control strategy for rumor control. Yue and Huo [30] developed a stochastic SICR rumor spreading model that accounted for media coverage and science education; the study’s results demonstrated that media coverage could inhibit rumor spread. Jain et al. [31] proposed the S ₁ S ₂ I model with expert interaction on a homogeneous network and discovered that noise perturbation caused persistent rumor spread. Huo and Dong [32] developed a stochastic ISR model with white noise media coverage, and the results of their study demonstrated that media coverage was inversely proportional to rumor spread. In addition, Mena et al. [33] analyzed the spread characteristics by developing an optimal control model based on the analysis of extinction and persistence of uncertain spread models.

The aforementioned scholars have conducted numerous studies on the influence of random factors on disease spread and rumor spread. In contrast, not all information is harmful to society in the real world, and positive information must be disseminated. Some beneficial information to society may be disseminated by individuals with authority, also known as super-spreaders. Then, it is more likely that the exposers who are in contact with the super-spreaders will comprehend and continue to spread the information. Conversely, the exposers who are not in contact with the super-spreaders are less likely to do so. The former is referred to as explicit spreaders, whereas the latter is referred to as implicit spreaders. This paper develops a stochastic S2EIR model that takes super-spreaders and exposers into account. Afterward, the existence of global positive solutions is demonstrated, suitable parameters are selected as control variables after calculating sufficient conditions for information disappearance and smooth information distribution, and the validity of the proposed theorem is validated through numerical simulation. Different from previous studies, this article considers that individuals with strong influence have played a significant role in promoting information spreading while also considering that some individuals are unwilling to spread information even if they are aware of it. Therefore, the common phenomena of “super-spreader” and “asymptomatic infection” in virus spreading are applied to the information spreading model. In addition, for each individual, the contact rate with information spreaders and the spread rate of being an information spreader may be affected by random perturbations in the environment. So, it is more practical to add random factors to the information spreading model. Meanwhile, the optimal control strategy containing random parameters of this paper is based on the optimal values calculated by the control variables.

The remainder of the paper is organized as follows. In Section 2, the stochastic S2EIR model taking super-spreaders and exposers into account is constructed. Section 3 proves the existence of a global positive solution. In Section 4, sufficient conditions for the disappearance of information are outlined. Section 5 provides sufficient conditions for stationary information distribution. The existence of optimal control and the optimal control strategy are discussed in Section 6. In Section 7, numerical simulation is used to analyze the impact of stochastic disturbance intensity on information spread and the effect of the control strategy. In the last section, conclusions are provided.

This paper examines an open virtual community whose population size fluctuates over time t. The total population size can be represented by N(t), where all populations can be divided into five categories: 1) the receptive population S(t) who do not have access to information but are receptive to it; 2) the explicitly exposed population E ₁(t) that has contact with the super-spreaders; 3) the implicitly exposed population E ₂(t) that does not have contact with the super-spreaders; 4) the spread-information population I(t); and 5) the non-receptive population R(t) that is no longer interested in the information.

The model flow diagram is presented in Figure 1.

In the model of this paper, the implicit exposers are rather special. People who are not influenced by the information’s super-spreaders typically do not easily comprehend the information’s content since it is spread by super-spreaders. Then, these people demonstrate a lack of interest in the information. Therefore, despite being exposed to the information, these individuals choose not to spread it.

The parameters in the S2EIR model can be interpreted as follows:

1) The number of individuals in a social system fluctuates over time. Therefore, in this paper, the author defines B as the number of individuals moving into the social system and defines μ as the emigration rate that is out of the social system due to force majeure factors.

In addition, the uncertain factors in social systems are commonly referred to as environmental noise. It is not scientific to study the spread of information while ignoring random environmental noise fluctuations. Incorporating environmental noise into deterministic models is more representative of how information spreads in real society. The random factors added to the spread models mainly include three classical approaches: 1) Introducing Gaussian white noise into deterministic parameter perturbation models [34]. 2) Random perturbation encompassing the positive endemic equilibrium of deterministic models [35]. 3) Alternating between regimes based on the probability of Markov chains [36]. Since random perturbations in the environment may affect the contact rate of each individual with the information spreader and the spread rate of becoming an information spreader, this paper uses Gaussian white noise to generate random perturbations of α and β, and the parameters of random perturbation are expressed as follows: α → α + σ 1 W ̇ 1 t , β → β + σ 2 W ̇ 2 t . (1)

Here, W _i (i = 1, 2) are independent standard Brownian motions and σ i 2 > 0 ( i = 1,2 ) represent the intensities of W _i (i = 1, 2), respectively. In this paper, W ₁ and W ₂ represent the relationship without mutual influence between α and β, respectively. In other words, in real scenarios, the conversion rate of information disseminators may be influenced by individual understanding and information acceptability, but it is not affected by the natural contact rate, and there is a state of mutual independence between the two.

The stochastic perturbation parameters are introduced into the deterministic model to construct a stochastic S2EIR model driven by Gaussian white noise, and the stochastic model can be represented as follows: d S t = B − α 1 + m S I − α S I − μ S d t − σ 1 1 + m S I d W 1 t − σ 1 S I d W 1 t , d E 1 t = α 1 + m S I − β E 1 − γ 1 E 1 − ε E 1 − μ E 1 d t + σ 1 1 + m S I d W 1 t − σ 2 E 1 d W 2 t , d E 2 t = α S I + ε E 1 − γ 2 E 2 − μ E 2 d t + σ 1 S I d W 1 t , d I t = β E 1 − λ I − μ I d t + σ 2 E 1 d W 2 t . (2)

In the rest of this paper, let ( Ω , F , { F t } t ≥ 0 , P ) be a complete probability space with a filtration { F t } t ≥ 0 satisfying the usual conditions. In addition, while F 0 contains all P − null sets, it is increasing and right continuous [37]. It also can be denoted as follows: R + 4 = x 1 , x 2 , x 3 , x 4 | x i > 0 , i = 1,2,3,4 . (3)

The existence of the global solution is the basis of analyzing the dynamic behavior of the stochastic system (Eq. 2). At the same time, according to the actual situation, a positive value is required for the dynamic model of information transmission. The stochastic system (Eq. 2) can be proved as global and positive by Theorem 3.1.

Theorem 3.1

The existence of a unique positive solution ( S ( t ) , E 1 ( t ) , E 2 ( t ) , I ( t ) ) ∈ R + 4 of the stochastic system (Eq. 2) is satisfied by any given initial value ( S ( t ) , E 1 ( t ) , E 2 ( t ) , I ( t ) ) ∈ R + 4 . The probability of the solution is 1 and remains in R + 4 .

Proof

The existence of a unique local positive solution ( S ( t ) , E 1 ( t ) , E 2 ( t ) , I ( t ) ) ∈ R + 4 of the stochastic system (Eq. 2) on t ∈ [0, τ _e ), which is based on the coefficients of the deterministic system, is locally Lipschitz continuous of any given initial value ( S ( 0 ) , E 1 ( 0 ) , E 2 ( 0 ) , I ( 0 ) ) ∈ R + 4 . τ _e is the explosion time [38]. It is required to have τ _e = ∞ a.s. to show this solution globally. The stopping time τ ⁺ can be defined by τ + = inf t ∈ 0 , τ e : S t ≥ 0 o r E 1 t ≥ 0 o r E 2 t ≥ 0 o r I t ≥ 0 . (4)

Let set inf ∅ = ∞ (∅ denotes the empty set). It is easy to get τ ⁺ ≤ τ _e . So, if τ ⁺ = ∞ a.s. is proved, then τ _e = ∞ and ( S ( t ) , E 1 ( t ) , E 2 ( t ) , I ( t ) ) ∈ R + 4 a.s. for all t ≥ 0. We assume that τ ⁺ < ∞; then, T > 0 is existence such that P(τ ⁺ < T) > 0. We define C 2 function V: R + 4 → R + 4 by V(X) = InSE ₁ E ₂ I. Using the I t o ̂ ′ s formula, we calculate the differential of V along the solution trajectories of the stochastic system (2). For ω ∈ (τ ⁺ < T) and for all t ∈ [0, τ _e ), we get d V X t = B S − α 1 + m I − α I − μ − 1 2 σ 1 2 1 + m 2 I 2 − 1 2 σ 1 2 I 2 d t + α 1 + m S I E 1 − β + γ 1 + ε + μ − 1 2 σ 1 2 1 + m 2 S 2 I 2 E 1 2 − 1 2 σ 2 2 d t + α S I E 2 + ε E 1 E 2 − γ 2 + μ − 1 2 σ 1 2 S 2 I 2 E 2 2 d t + β E 1 I − λ + μ − 1 2 σ 2 2 E 1 2 I 2 d t − σ 1 1 + m I d W 1 − σ 1 I d W 1 + σ 1 1 + m S I E 1 d W 1 − σ 2 d W 2 + σ 1 S I E 2 d W 1 + σ 2 E 1 I d W 2 . (5)

Positivity of X(t) implies that d V X t ≥ L S , E 1 , E 2 , I d t − σ 1 1 + m I + I − 1 + m S I E 1 − S I E 2 d W 1 − σ 2 1 − E 1 I d W 2 , (6) where L S , E 1 , E 2 , I = − μ − β + γ 1 + ε + μ − γ 2 + μ − λ + μ − 1 2 σ 1 2 1 + m 2 I 2 − 1 2 σ 1 2 I 2 − 1 2 σ 1 2 1 + m 2 S 2 I 2 E 2 2 − 1 2 σ 1 2 S 2 I 2 E 2 2 − 1 2 σ 2 2 − 1 2 σ 2 2 E 1 2 I 2 . (7)

So, we have V X t ≥ V X 0 + ∫ 0 t L S u , E 1 u , E 2 u , I u d u − ∫ 0 t σ 1 I u − 1 + m S u I u E 1 u − S u I u E 2 u d W 1 u − ∫ 0 t σ 2 1 − E 1 u I u d W 2 u . (8)

Note that some components of X(τ ⁺) equal 0. Thereby, lim t → τ + V X t = − ∞ . (9)

Letting t → τ ⁺ in the system (Eq. 8), we have − ∞ ≥ V X 0 + ∫ 0 τ + L S u , E 1 u , E 2 u , I u d u − ∫ 0 τ + σ 1 I u − 1 + m S u I u E 1 u − S u I u E 2 u d W 1 u − ∫ 0 τ + σ 2 1 − E 1 u I u d W 2 u > − ∞ . (10)

According to Eqs. 8–9, it can be obtained that Eq. 10 is less than or equal to −∞. Meanwhile, for any given initial value, ( S ( 0 ) , E 1 ( 0 ) , E 2 ( 0 ) , I ( 0 ) ) ∈ R + 4 and S(u), E ₁(u), E ₂(u), I(u) in Eq. 10 belong to a positive invariant set and are bounded. Therefore, S(u), E ₁(u), E ₂(u), I(u) are greater than 0 and greater than −∞, and then, Eq. 10 is greater than −∞. This result is contradictory. In addition, the result obtained by Eq. 10 rejects the original hypothesis τ ⁺ < ∞. Thus, τ ⁺ = ∞.

Theorems 4.1 and 4.2 give the condition for the disappearance of the information. The condition is expressed by intensities of noises and parameters of the deterministic system. In the stochastic S2EIR model built in this paper, information disappearance needs to meet the following conditions: 1) all exposers affected by the super-spreaders disappeared, and 2) all spreaders of information disappeared. If any of the two aforestated conditions is satisfied, information can disappear in the social system.

First, Theorem 4.1 gives the condition for the disappearance of information caused by the disappearance of the exposers affected by the super-spreaders.

Theorem 4.1

: For any given initial value ( S ( 0 ) , E 1 ( 0 ) , E 2 ( 0 ) , I ( 0 ) ) ∈ R + 4 , lim t → ∞ sup ln E 1 ( t ) t ≤ G ( σ 1 2 , σ 2 2 ) holds a.s. Furthermore, G ( σ 1 2 , σ 2 2 ) < 0 . Then, E ₁(t) tend to 0 exponentially a.s., where G ( σ 1 2 , σ 2 2 ) = α 2 2 σ 1 2 − ( β + γ 1 + ε + μ + 1 2 σ 2 2 ) .

Proof

use the I t o ̂ ′ s formula to calculate the differentiation of E ₁(t) in the stochastic system (2), and d ln E ₁(t) can be written as d ⁡ ln E 1 t = α 1 + m S I E 1 − β + γ 1 + ε + μ − 1 2 σ 1 2 1 + m 2 S 2 I 2 E 1 2 − 1 2 σ 2 2 d t + σ 1 1 + m S I E 1 d W 1 − σ 2 d W 2 . (11)

Thus, ln E ₁(t) can be denoted as ln E 1 t = ln E 1 0 + ∫ 0 t α 1 + m S u I u E 1 u − β + γ 1 + ε + μ − 1 2 σ 1 2 1 + m 2 S 2 u I 2 u E 1 2 u − 1 2 σ 2 2 d u + ∫ 0 t σ 1 1 + m S u I u E 1 u d W 1 u − σ 2 W 2 t . (12) Φ 1 t = ∫ 0 t σ 1 1 + m S u I u E 1 u d W 1 u , (13)

Here, Φ₁(t) is a continuous local martingale. The quadratic variation of Φ₁(t) can be denoted as Φ 1 t = σ 1 2 ∫ 0 t 1 + m 2 S 2 u I 2 u E 1 2 u d u . (14)

By exponential martingale inequality [38], it can be known that P sup 0 ≤ t ≤ k Φ t − c 2 Φ t > 2 c ln ⁡ k ≤ k − 2 c , (15) where 0 < c < 1, k is a random integer. Using the Borel–Cantelli lemma, it is easy to know that the random integer k ₀(ω) exists such that for k > k ₀, for almost all ω ∈ Ω, sup 0 ≤ t ≤ k [ Φ ( t ) − c 2 Φ ( t ) ] ≤ 2 c . Therefore, for all t ∈ [0, k], we have ∫ 0 t σ 1 1 + m S u I u E 1 u d W 1 u ≤ 1 2 c σ 1 2 ∫ 0 t 1 + m 2 S 2 u I 2 u E 1 2 u d u + 2 c ln ⁡ k . (16)

Then, it can be obtained that ln E 1 t ≤ ln E 1 0 + ∫ 0 t α 1 + m S u I u E 1 u − β + γ 1 + ε + μ − 1 2 1 − c σ 1 2 1 + m 2 S 2 u I 2 u E 1 2 u − 1 2 σ 2 2 d u + 2 c ln ⁡ k − σ 2 W 2 t , (17) noting that α 1 + m S u I u E 1 u − 1 2 1 − c σ 1 2 1 + m 2 S 2 u I 2 u E 1 2 u ≤ α 2 2 1 − c σ 1 2 . (18)

Substituting Eq. 18 into Eq. 17, ln E ₁(t) can be written as ln E 1 t ≤ ln E 1 0 + ∫ 0 t α 2 2 1 − c σ 1 2 − β + γ 1 + ε + μ − 1 2 σ 2 2 d u + 2 c ln ⁡ k − σ 2 W 2 t = ln E 1 0 + α 2 2 1 − c σ 1 2 − β + γ 1 + ε + μ + 1 2 σ 2 2 t + 2 c ln ⁡ k − σ 2 W 2 t . (19)

Hence, for k − 1 ≤ t ≤ k, ln E 1 ( t ) t can be obtained as ln E 1 t t ≤ ln E 1 0 t + α 2 2 1 − c σ 1 2 − β + γ 1 + ε + μ + 1 2 σ 2 2 + 2 c ln ⁡ k − σ 2 W 2 t . (20)

By the strong law of large numbers to the Brownian motion, let k → ∞ and then t → ∞. It can be known that lim t → ∞ sup W 2 ( t ) t = 0 .

Therefore, lim t → ∞ sup ln E 1 t t ≤ α 2 2 1 − c σ 1 2 − β + γ 1 + ε + μ + 1 2 σ 2 2 . (21)

Finally, let c → 0. Then, lim t → ∞ sup ln E 1 ( t ) t can be obtained as lim t → ∞ sup ln E 1 t t ≤ α 2 2 σ 1 2 − β + γ 1 + ε + μ + 1 2 σ 2 2 . (22)

Next, Theorem 4.2 gives the condition for the disappearance of information caused by the disappearance of the spreaders.

Theorem 4.2

For any given initial value, ( S ( 0 ) , E 1 ( 0 ) , E 2 ( 0 ) , I ( 0 ) ) ∈ R + 4 , lim t → ∞ sup ln ⁡ I ( t ) t ≤ G ( σ 2 2 ) holds a.s. Furthermore, G ( σ 2 2 ) < 0 . Then, I(t) tend to 0 exponentially a.s., where G ( σ 2 2 ) = β 2 2 σ 2 2 − ( λ + μ ) .

Proof

use the I t o ̂ ′ s formula to calculate the differentiation of I(t) in the stochastic system (Eq. 2), and d ln I(t) can be written as d ⁡ ln ⁡ I t = β E 1 I − λ + μ − 1 2 σ 2 2 E 1 2 I 2 d t + σ 2 E 1 I d W 2 . (23)

Thus, ln I(t) can be denoted as ln ⁡ I t = ln ⁡ I 0 + ∫ 0 t β E 1 u I u − λ + μ − 1 2 σ 2 2 E 1 2 u I 2 u d u + ∫ 0 t σ 2 E 1 u I u d W 2 u . (24)

We denote Φ 2 t = ∫ 0 t σ 2 E 1 u I u d W 2 u , (25) where Φ₂(t) is a continuous local martingale. The quadratic variation of Φ₂(t) can be denoted as Φ 2 t = σ 2 2 ∫ 0 t E 1 2 u I 2 u d u . (26)

Similar to Theorem 4.1, for all t ∈ [0, k], one can obtain ∫ 0 t σ 2 E 1 u I u d W 2 u ≤ 1 2 c σ 2 2 ∫ 0 t E 1 2 u I 2 u d u + 2 c ln ⁡ k . (27)

Then, it can be obtained that ln ⁡ I t ≤ ln ⁡ I 0 + ∫ 0 t β E 1 u I u − λ + μ − 1 2 1 − c σ 2 2 E 1 2 u I 2 u d u + 2 c ln ⁡ k . (28)

It can be noted that β E 1 u I u − 1 2 1 − c σ 2 2 E 1 2 u I 2 u ≤ β 2 2 1 − c σ 2 2 . (29)

Substituting Eq. 29 into Eq. 28, ln I(t) can be written as ln ⁡ I t ≤ ln ⁡ I 0 + ∫ 0 t β 2 2 1 − c σ 2 2 − λ + μ d u + 2 c ln ⁡ k = ln ⁡ I 0 + β 2 2 1 − c σ 2 2 − λ + μ t + 2 c ln ⁡ k . (30)

Hence, for k − 1 ≤ t ≤ k, ln ⁡ I ( t ) t can be obtained as ln ⁡ I t t ≤ ln ⁡ I 0 t + β 2 2 1 − c σ 2 2 − λ + μ + 2 c ⋅ ln ⁡ k k − 1 . (31)

By the strong law of large numbers to the Brownian motion, let k → ∞ and then t → ∞. It can be known that lim t → ∞ sup ln ⁡ I t t ≤ β 2 2 1 − c σ 2 2 − λ + μ . (32)

Finally, let c → 0. Then, lim t → ∞ sup ln ⁡ I ( t ) t can be obtained as lim t → ∞ sup ln ⁡ I t t ≤ β 2 2 σ 2 2 − λ + μ . (33)

Remark 4.1: G ( σ 1 2 , σ 2 2 ) = α 2 2 σ 1 2 − ( β + γ 1 + ε + μ + 1 2 σ 2 2 ) and G ( σ 2 2 ) = β 2 2 σ 2 2 − ( λ + μ ) are decreasing in σ 1 2 and σ 2 2 . The information will disappear eventually if σ 1 2 and σ 2 2 are large enough, where G ( σ 1 2 , σ 2 2 ) < 0 and G ( σ 2 2 ) < 0 .

Theorem 5.1 gives the unique stationary distribution of the existence of the stochastic system (Eq. 2). This also means stability in a stochastic sense.

Theorem 5.1

If R 0 = B α β ( 1 + m ) μ ( β + γ 1 + ε + μ ) ( λ + μ ) > 1 , the stochastic system (Eq. 2) with initial condition ( S ( 0 ) , E 1 ( 0 ) , E 2 ( 0 ) , I ( 0 ) ) ∈ R + 4 and the following conditions are satisfied: 0 < Γ < min ξ 1 S 2 , ξ 2 E 1 2 , ξ 3 E 2 2 , ξ 4 I 2 , (34) where Γ = 2 σ 1 2 S * 2 I * 2 + 1 2 σ 2 2 E 1 * + σ 2 2 E 1 * 2 , ξ 1 = μ , ξ 2 = γ 1 + μ − σ 2 2 , ξ 3 = γ 2 + μ , ξ 4 = λ + μ . (35)

Then, the stationary distribution π exists, and the solution of the stochastic system (Eq. 2) is ergodic.

By the information-existence equilibrium point E * = ( S * , E 1 * , E 2 * , I * ) , where S * = B μ R 0 2 , E 1 * = B ( 1 + m ) ( R 0 2 − 1 ) ( 2 + m ) ( β + γ 1 + ε + μ ) R 0 2 , E 2 * = B ( β + γ 1 + ε + μ ) ( R 0 2 − 1 ) + B ε ( 1 + m ) ( R 0 2 − 1 ) ( 2 + m ) ( γ 2 + μ ) ( β + γ 1 + ε + μ ) R 0 2 , and I * = μ ( R 0 2 − 1 ) α ( 2 + m ) . Then, it can be obtained that lim t → ∞ 1 t E ∫ 0 t ξ 1 S u − S * 2 + ξ 2 E 1 u − E 1 * 2 + ξ 3 E 2 u − E 2 * 2 + ξ 4 I u − I * 2 d u < Γ . (36)

Proof

define a C 2 function V: Θ S , E 1 , E 2 , I = Θ 1 S + Θ 2 E 1 + Θ 3 E 2 + Θ 4 I + Θ 5 S , E 1 , E 2 , I , (37) where Θ 1 E 1 = E 1 − E 1 * − E 1 * ⁡ ln E 1 E 1 * , Θ 2 E 2 = E 2 − E 2 * − E 2 * ⁡ ln E 2 E 2 * , Θ 3 I = I − I * − I * ⁡ ln I I * , Θ 4 S , E 1 , E 2 , I = 1 2 S + E 1 + E 2 + I − S * − E 1 * − E 2 * − I * 2 . (38)

The differential L operator to Θ₁ can be calculated as L Θ 1 = α 1 + m S I − β + γ 1 + ε + μ E 1 ∂ Θ 1 ∂ E 1 + 1 2 σ 1 2 1 + m 2 S 2 I 2 + σ 2 2 E 1 2 ∂ 2 Θ 1 ∂ E 1 2 = E 1 − E 1 ∗ α 1 + m S I E 1 − β + γ 1 + ε + μ + 1 2 σ 1 2 1 + m 2 S 2 I 2 + 1 2 σ 2 2 E 1 ∗ . (39)

According to E * = ( S * , E 1 * , E 2 * , I * ) , it is easy to get that β + γ 1 + ε + μ = α 1 + m S * I * E 1 * , (40) and then, LΘ₁ can be expressed as L Θ 1 = E 1 − E 1 ∗ α 1 + m S I E 1 − S * I * E 1 * + 1 2 σ 1 2 1 + m 2 S 2 I 2 + 1 2 σ 2 2 E 1 ∗ = E 1 − E 1 ∗ − α 1 + m S I E 1 − E 1 ∗ E 1 E 1 ∗ + α 1 + m S − S * I − I * E 1 ∗ + 1 2 σ 1 2 1 + m 2 S 2 I 2 + 1 2 σ 2 2 E 1 ∗ , (41) where α ( 1 + m ) S I ( E 1 − E 1 * ) 2 E 1 E 1 * ≥ 0 and E 1 * > 0 .

By simple calculation, one can get L Θ 1 ≤ α 1 + m S − S * I − I * E 1 − E 1 * + 1 2 σ 1 2 1 + m 2 S − S * + S * 2 I − I * + I * 2 + 1 2 σ 2 2 E 1 * . (42)

Due to 1 2 ( x + y ) 2 ≤ x 2 + y 2 , it is easy to obtain that L Θ 1 ≤ α 1 + m S − S * I − I * E 1 − E 1 * + σ 1 2 1 + m 2 S − S * 2 I − I * 2 + σ 1 2 S * 2 I * 2 + 1 2 σ 2 2 E 1 * . (43)

Similarly, the differential L operator to Θ₂ can be calculated as L Θ 2 = α S I + ε E 1 − γ 2 E 2 − μ E 2 ∂ Θ 2 ∂ E 2 + 1 2 σ 1 2 S 2 I 2 ∂ 2 Θ 2 ∂ E 2 2 = E 2 − E 2 * α S I E 2 + ε E 1 E 2 − γ 2 + μ + 1 2 σ 1 2 S 2 I 2 . (44)

According to E * = ( S * , E 1 * , E 2 * , I * ) , it is easy to get that γ 2 + μ = α S * I * E 2 * + ε E 1 * E 2 * , (45) and then, LΘ₂ can be expressed as L Θ 2 = E 2 − E 2 * α S I E 2 − α S * I * E 2 * + ε E 1 E 2 − ε E 1 ∗ E 2 * + 1 2 σ 1 2 S 2 I 2 = E 2 − E 2 ∗ − α S I E 2 − E 2 ∗ E 2 E 2 ∗ − ε E 1 E 2 − E 2 ∗ E 2 E 2 * + α S − S * I − I * E 2 * + ε E 1 − E 1 * E 2 * + 1 2 σ 1 2 S 2 I 2 , (46) where α S I ( E 2 − E 2 * ) 2 E 2 E 2 * ≥ 0 , ε E 1 ( E 2 − E 2 * ) 2 E 2 E 2 * ≥ 0 , and E 2 * > 0 .